In Support of Trivialism

I agree that “Tom is handsome” means “Tom’s appearance is handsome,” but that does not defeat my point. Now, instead of having handsome be an incomplete descriptor for Tom, you have handsome as an incomplete descriptor for Tom’s appearance. Tom’s appearance does not equal handsome. Handsome is a general, abstract quality that other peoples’ appearances sometimes have and that some peoples’ appearances have in theory or fiction. There may be other men in addition to Tom, such as Caleb and Fabian, whose appearances are also handsome. Handsome, alone, does not completely, exclusively, and uniquely describe Tom’s appearance.

As I had explained earlier in this thread in my posts at viewtopic.php?p=2695954#p2695954, there are multiple senses of equality. Two rectangles can be equal in one sense, but unequal in another sense. The word is and the equality symbol = are not as straightforward in their meanings as you may think they are.

Yes, it is. It’s an argument that has cited and is backed by a well respected source.

Rosen’s assertion is good evidence that all propositions are sentences.

Rosen and I both agree that not all sentences are propositions.

Rosen has claimed that a sentence can be a proposition. They can be the same thing. While I am surprised by that idea, I do not object to it. If you do, perhaps you could give some counterexamples of propositions that are not sentences.

I did say that, in my original post, at viewtopic.php?p=2695519#p2695519, first sentence of the third paragraph.

The statement “a rectangle is a square” is a statement of basic, common English. Statements like that, including “a quadrilateral is a parallelogram,” “a cucumber is green,” “one angle is congruent to a second angle,” and “two lines are parallel,” are statements of basic, common English. They are used in high school geometry textbooks here in the United States, including the aforementioned Geometry (2004) and Larson Geometry (2012). In some of my previous discussions involving my argument for trivialism, I have cited some grade school geometry textbooks, other than the aforementioned two, that use simple statements of the described type. Those discussions are available through a link I have provided in the original post for this thread.

I quote you from ilovephilosophy.com/viewtopic.ph … 0#p2697559, which you previously cited.

If trivialism is true, then a light can be on and not on simultaneously.

I used to have a postulate in my Action-Reaction Theory called the Postulate of Temporal Extensionality. There’s a video about the postulate that involves a light being simultaneously on and off; it’s at facebook.com/Paul.E.Mokrzec … 008356517/.

Thank you

This is interesting but I’m inclined to believe there is more unspoken meaning embedded in the phrase because one attribute of Tom simply must be equal to handsome; we just haven’t decided exactly what it is.

I agree that Tom may also be tall, or fat, or whatever in addition to handsome so it is therefore not a complete descriptor of his appearance, but on the other hand “handsome” is a complete descriptor of something. If that is true, then once again we find we haven’t constructed the sentence properly to accurately represent the proposition.

“Tom’s appearance is handsome.” That doesn’t work because the statement is objective while “handsome” is subjective.

“Sally finds Tom’s appearance is handsome.” Now we’ve aligned the subjectivity, but maybe Tom is not always handsome.

“Throughout the day, Sally finds Tom’s appearance is handsome.” Now we’re aligned temporally, but can Tom be perfectly handsome?

“Throughout the day, Sally finds Tom’s appearance is consistent with what she categorically regards as handsome.” On and on until it seems like a police report as we zero-in on exactly what is being equivocated.

Two rectangles being equal in one sense and not another is just another example of what was stated above: there are embedded meanings that need to be unpackaged. What is equivalent between all rectangles is having 4 parallel sides and right angles, so it’s not the rectangles that are equal, but aspects of them. If you want to state equivalence beyond that, then the rectangles in fact must be the same size.

Are you aware of instances where respected sources have been wrong? If so, then how can authority ever be a definitive authority?

An assertion is not evidence. Evidence backs the assertion.

What sentences are not propositions? Questions? Statements state propositions and questions question them.

I gave you an example: a raft used for crossing a river is not the same as the passengers; it is just a means of getting across the river; a conveyance. Likewise, a sentence is not the proposition, but carries the proposition from one head to another head.

You said “Some rectangles are squares. So, a rectangle is a square.” You cannot make that leap. Alternatively, you could have said “Some rectangles are squares. So, a specific type of rectangle is a square.”

They are slang.

The color of the skin of specific varieties of ripened, but not over-ripened cucumbers are often green. Instead of saying all that, we simply say cucumbers are green, but it’s not a precise statement. You cannot take advantage of linguistic conveniences to prove something about reality.

It is either unfortunate that those textbooks use such language or the books rely on the student’s ability to interpolate properly.

That assumes that all events that have ever happened or ever will happen actually exist, right now.

Start at about 4:00

[youtube]https://www.youtube.com/watch?v=MO_Q_f1WgQI[/youtube]

Then toss in Many Worlds where all possibilities also exist, then we can say that everything and anything exists right now.

Now what?

Little Fluffy got hit by a car? Oh don’t fret, there are infinite Fuffies still alive somewhere.

No, your honor, I didn’t steal the money. The money is still there in the past and in multiple universes. There is no crime.

I suppose this means I haven’t been born yet. Oh good! There are a few things I’d like to do differently :wink:

I believe the books are presenting the material correctly. So correctly, in fact, that I treat their presentation like a standard in my analysis.

Both instances of the noun phrase “a rectangle” there do not have to refer to the same rectangle. They simply have to refer to a rectangle. What rectangle or rectangles they refer to does not matter. Whether they refer to the same rectangle or to different rectangles does not matter.

Yes but that’s just your opinion.

The category of “rectangle” is different from a specific rectangle. You’re conflating/equivocating categories with members of categories.

“A rectangle is a square” should be changed to “a (specific) rectangle is (in the category of) a square”.

You can’t say the category of rectangle is the same as the category of square nor that a specific rectangle is necessarily the same as a specific square.

I know that.

Some rectangles are squares. So, a specific rectangle is in the category of square. Some rectangles are not squares. So, a specific rectangle is not in the category of square. It follows by conjunction introduction that “a specific rectangle is in the category of square and a specific rectangle is not in the category of square.” But the quoted statement is a contradiction. Therefore, by the principle of explosion, trivialism is true.

Not quite.

“a specific rectangle (R1) is in the category of square and a specific rectangle (R2) is not in the category of square.” No contradiction.

Serendipper:

Whether each instance of “a specific rectangle” refers to the same specific rectangle or to different specific rectangles does not matter. Each instance simply refers to a specific rectangle.

If each instance refers to a specific rectangle, then in that instance, the specific rectangle will either be a square or not. Still no contradiction.

It’s like your example with the lamps: the lamp was on in the instance in the past and off in the instance in the present.

You may have some luck in the quantum realm where two particles can be in different states at the same time, but the particles could be waves rather than particles so there’s some debate even then.

Why are you so determined to prove trivialism anyway?

I agree.

There is still a contradiction because

is still true.

Proving trivialism gives us a proof for every claim that can possibly be made. No more proofs will ever be required. Instead of proving claims with possibly long, complicated, and tedious proofs, academics, scientists, and others can advance to other things.

It can only be a contradiction if it is the same rectangle that is and is not a square, which it cannot be by definition of the proposition.

Here is an image I found online

Members of the area labeled 5.1 do not mingle with members of the area labeled 3.4. The area 3.4 represents squares and 5.1+3.4 represents rectangles. You may as well be saying 3.4 = 5.1+3.4

Well then I claim trivialism is false :wink:

The subject of the statement “a specific rectangle is in the category of square” and the subject of the statement “a specific rectangle is not in the category of square” are the same. That common subject is “a specific rectangle.” Thus, “a specific rectangle is not in the category of square” is the negation of “a specific rectangle is in the category of square.” Since, according to the modified argument, both are true, a contradiction exists.

The subjects are the same in at least one sense. This is like how there are two separate subjects in the sense that the subjects are of two separate statements, but there is exactly one subject in the sense that the subject of both statements is “a specific rectangle.” Since the subjects are the same in at least one sense, “a specific rectangle is not in the category of square” is the negation of “a specific rectangle is in the category of square” in at least one sense. Since, according to the modified argument, both are true, a contradiction exists in at least one sense. So, a contradiction exists. Therefore, by the principle of explosion, trivialism is true.

I’ve already acknowledged, in a reply to fuse at viewtopic.php?p=2695577#p2695577, that if trivialism is true, then trivialism is also false. Since trivialism is true, I agree with you that trivialism is false. Nonetheless, trivialism is true.

No.

R1 not equal to R2. They are defined to be different by your proposition that one is a square and the other is not. You cannot define a contradiction to exist and then claim that means something.

No contradiction.

Irrelevant.

They are separate senses.

No

Nope

If it’s false it can’t be true. This is the silliest way to use time.

All you’re doing is claiming: X contradicts Y because you said so; therefore Z is true and because it’s true, it’s also false. It’s a strawman. You set it up just to knock it down and then claim it means something.

I suppose we can keep going back n forth like school children: no huh uh, yes huh, no huh uh, yes huh, until one of us gets bored with it. It’s slow around ILP now, so let’s see what happens I guess.

I have a proposal that might help you browser32… attempt to present your deductive argument using symbolic logic.

You’ll find that it’s very hard to mask the sophistry without the ability to equivocate using ambiguous language…

R1 may not be equal to R2 in some sense of equality. But in some sense of equality, R1 is equal to R2 because each refers to a specific rectangle. R1 = a specific rectangle. R2 = a specific rectangle. Therefore, by the substitution property of equality, R1 = R2.

R1 and R2 share a property in common with each other, and that common property alone can be used to equate the two objects in some sense of equality. R1 and R2 are equal in merely the sense that they share the same name. That name is “a specific rectangle.”

Example. Let b be a United States dime and f be a United States dime. b does not equal f in some sense of equality because b was manufactured in 2003, but f was manufactured in 2007. However, b is equal to f in the sense that both share the property of being a United States dime. Furthermore, b is equal to f in the sense that both have the same monetary value, 10 cents. This concludes the example.

The existence of different senses of identity like in the example was discussed in a course I took my first semester of college called Minds and Machines. I did not graduate from college, but according to an unofficial transcript, I did get an A- in the course.

I agree, and I am not conflating senses there. There, I am explaining how there are at least two senses.

The following is an excerpt from my reply to phyllo at viewtopic.php?p=2695577#p2695577.

Mad Man P:

I don’t believe my argument can be presented in first-order logic; at least not in a traditional way as I envision it. It is somewhat bold for one to think that all arguments can be presented in commonly accepted, formal, symbolic logic. The existence of second-order logic and other logical systems suggest that not all arguments can be presented in first-order logic. See the first two sentences of en.wikipedia.org/wiki/Second-or … sive_power. My argument is presented in the language of symbolic logic known as English.

We previously talked about the logical rule of inference known as existential instantiation. With the rule of existential instantiation, I would be required to give the rectangle referred to by “a rectangle is not a square” a unique name that is different from the name I give the rectangle referred to by “a rectangle is a square.” One potential flaw with existential instantiation is that it seems to be based off of the implicit assumption that two things that are unequal are not permitted to have the same name. In mathematics in general, that assumption seems to be made. However, in the real world, two things that are unequal can, and sometimes do, have the same name. So, it seems that traditional mathematics is flawed because it seems to have an unnecessary rule that reality does not abide by.

Conflation among different things with the same name may be an inevitable or necessary feature of nature.

Nonetheless, for the sake of advancement here, I make the following attempts to present my argument in symbolic logic.

Argument 1.
Domain: All rectangles.
[i]R/i is the statement “x is a rectangle.”
[i]S/i is the statement “x is a square.”
P is a statement.
Statements (Reasons)

  1. x([i]R/i ˄ [i]S/i) (Premise)
  2. [i]R/i ˄ [i]S/i (Existential instantiation from (1))
  3. x([i]R/i ˄ ¬[i]S/i) (Premise)
  4. [i]R/i ˄ ¬[i]S/i (Existential instantiation from (3))
  5. d = g (Premise)
  6. [i]R/i ˄ ¬[i]S/i (Substitution property of equality from (4) and (5))
  7. [i]S/i (Conjunction elimination from (2))
  8. ¬[i]S/i (Conjunction elimination from (6))
  9. ꓕ (ꓕ introduction from (7) and (8))
  10. P (ꓕ elimination from (9))
    This concludes the argument.

The most questionable step in the above argument is (5), which I provided as a premise. The premise is that the rectangle that is a square is equal to the rectangle that is not a square. The premise’s truth is based off of the fact that d and g each have the same name, a rectangle. Notice that I did not indicate what was given and what was to be proved at the beginning of the argument. That was because premise (5) was not available at the beginning of the argument. Premise (5) involves and thus is dependent upon the context developed previously in the argument. The following argument integrates premise (5) and the other premises into a single premise.

Argument 2.
Domain: All rectangles.
[i]R/i is the statement “x is a rectangle.”
[i]S/i is the statement “x is a square.”
P is a statement.

Given: ꓱx(([i]R/i ˄ [i]S/i) ˄ ([i]R/i ˄ ¬[i]S/i))
Prove: P

Statements (Reasons)

  1. x(([i]R/i ˄ [i]S/i) ˄ ([i]R/i ˄ ¬[i]S/i)) (Given)
  2. ([i]R/i ˄ [i]S/i) ˄ ([i]R/i ˄ ¬[i]S/i) (Existential instantiation from (1))
  3. [i]R/i ˄ [i]S/i ˄ [i]R/i ˄ ¬[i]S/i (Simplification from (2))
  4. [i]S/i (Conjunction elimination from (3))
  5. ¬[i]S/i (Conjunction elimination from (3))
  6. ꓕ (ꓕ introduction from (4) and (5))
  7. P (ꓕ elimination from (6))
    This concludes the argument.

In the second argument, it is postulated that there is a rectangle that both is and is not a square. The basis for that contradiction is that the rectangle that is a square and the rectangle that is not a square are the same because each is a rectangle. They each have the property of being a rectangle. They each have the name a rectangle. They each share the noun phrase a rectangle.

No, R1 = specific rectangle and R2 = a different specific rectangle. That’s your definition in order to meet the requirement of one being a square and the other not. So now you can’t turn around and make a conclusion based on your definition: “because I defined them to be different, but labeled them the same, they are therefore equal and contradictory, therefore trivialism is true.”

You’ve taken different things by definition and labeled them the same then claimed a contradiction exists.

See? That’s exactly what you have done. So I could take a cat and dog, label the dog a cat, then claim a cat is equal to a dog because they have the same name, and then claim a contradiction.

I still maintain you are conflating senses. The value of a dime is equal to the value of another dime, but they are not the same dime.

A better example is a red block and blue block that are exactly the same other than color. We can say they are the same in one sense, but not in the other sense and therefore they are not equal except in their respective senses which you cannot conflate with arbitrary labeling.

I’m a human. The Pope is a human. Therefore I am the Pope.

Now go in peace and sin no more. You can cos as much as you like.

Sin and virtue are attributes of action and are therefore equal :smiley:

That’s right. If there exists a rectangle that’s a square, we symbolize this as

$$\exists x ( R(x) \land S(x))$$

We say that x is now a bound variable. As Wiki puts it:

That is, free variables become bound, and then in a sense retire from being available as stand-in values for other values in the creation of formulae.

en.wikipedia.org/wiki/Free_vari … _variables

So now if there’s some other rectangle that’s not a square, and we want to express this fact in conjunction with the earlier fact, we write

$$\exists x ( R(x) \land S(x)) \land \exists y (R(y) \land \neg S(y))$$

It’s not a flaw, it’s a feature. It’s how logic works. The purpose of symbolic logic is to be crystal clear in our meaning; to avoid the ambiguity of natural language. That’s the entire point.

Certainly we can use the same name for different objects in different contexts. But in formal logic, we first choose the context, or domain, or universe – different words for the same idea – and then within that context, names must refer uniquely to objects.

Well sure, a cat is a furry four-legged handwarmer; and a cat is a Caterpillar tractor; and a cat is a person with a hip demeanor, as in a cool cat.

What of it? Again, the entire point of symbolic logic is to remove the ambiguity of natural language, so that we may be sure that we are reasoning correctly. Of course natural language is ambiguous, that’s so poets will have something to do. The fog comes on little cat feet.

In poetry, we exploit the ambiguity of natural language; in formal logic, we avoid it. Poetry and logic. Two different human activities.

Math is good for doing math, and everyday natural language is good for doing everyday natural things. I don’t understand why you think one is “flawed.” We use hammers to hammer nails, and flashlights to illuminate the dark. We don’t say hammers are flawed because they’re not flashlights. We use different tools for different tasks. Surely you agree.

No, it’s an inevitable or necessary feature of natural language. You are confusing the names of things with the things themselves. You are confusing the words we use to talk about nature and to navigate the world; with the world itself.

This is a philosophical error. There are things, and there are names. The names of things are not the things.

I commend you for recognizing that this step is problematic.

But surely you see that just because I am a human and the Pope is a human, that I am not necessarily the Pope!

Here “is” does not mean “equals,” which would allow you to use transitivity of equality: if A = B and B = C then A = C.

Rather in this instance, “is” means, “is a member of some class.” So you have some horse that “is” an animal; and you have a cat that “is” an animal. But a horse is not a cat.

In this case “is” is being used as set membership, if you like; but not as equality.

Regardless of whether the rectangle referred to in “a rectangle is not a square” is the same rectangle referred to in “a rectangle is a square,” the former statement is the negation of the latter statement.

The following Argument 3 is a third attempt to present my argument in symbolic logic.

Argument 3.

Given: p = “A rectangle is a square,” p, ¬p, q is a statement.
Prove: q

Statements (Reasons)

  1. p (Given)
  2. ¬p (Given)
  3. ꓕ (ꓕ introduction from (1) and (2))
  4. q (ꓕ elimination from (3))
    This concludes the argument.

Premise (1) is justified by the fact that some rectangles are squares. Premise (2) is justified by the fact that some rectangles are not squares.

I know it sounds odd, but in a sense you are the Pope. You are the Pope in the sense that you and the Pope each is a human. There may be contexts in which you would use such discourse.

Example. You are talking to aliens that are not humans and are not from earth. You indicate to them that, among the numerous species of life on earth, you and the Pope are of the same species. You told the aliens you are not a cat, you are not a spider, you are not a giraffe, and you are not a penguin. You told the aliens, however, that you are the Pope. The aliens understand that you meant that there are some differences between you and the Pope, but that you and the Pope are of the same species. This concludes the example.

Again, there are multiple senses of equality. Since you are a human and the Pope is a human, there is a sense of equality in which “you = a human” and “the Pope = a human.” This sense of equality may be syntactic only, or it may be syntactic and semantic. You and the Pope have properties in common that can be used to pair you two up and regard you two in the same way.

Natural language is a part of nature.

It doesn’t matter what language a contradiction is asserted in; if a contradiction exists, then through the principle of explosion, all statements of all languages are true. Thus, if a contradiction exists in natural language, then by the principle of explosion, all statements of first-order logic, all statements of all natural languages, and all statements of all unnatural languages are true.

The names of things are not always the things. The names of things can be considered properties of the things. It actually seems the name of a thing is often considered a property of the thing. For example, in computer software, the name of an object is often considered one of the most important properties of the object.

Yes, and in biology a cat is a four legged furry handwarmer, while in popular culture it’s an especially hip hipster. What of it?

Regarding your equivocation of “is”, consider a mathematical example.

2 is a number and 3 is a number, but 2 is not equal to 3.

If we let N be the set of natural numbers, then we say 2 ∈ N and 3 ∈ N. That means “2 is a member of the set of natural numbers, and 3 is a member of the set of natural numbers.”

In this case the natural language “is” refers to set membership.

When we say that 2 = 1 + 1 and 1 + 1 = 5 - 3, that’s equality. It’s a transitive relation, so that we may conclude that 2 = 5 - 3.

You are simply equivocating “is” as set or class membership, and “is” as the equality relationship.